Properties

Label 2.139.abp_bat
Base field $\F_{139}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{139}$
Dimension:  $2$
L-polynomial:  $1 - 41 x + 695 x^{2} - 5699 x^{3} + 19321 x^{4}$
Frobenius angles:  $\pm0.105232638034$, $\pm0.208549921524$
Angle rank:  $2$ (numerical)
Number field:  4.0.2043717.2
Galois group:  $D_{4}$
Jacobians:  $16$

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:  $1$
Slopes:  $[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $14277$ $367732689$ $7211121088383$ $139362633210982077$ $2692477571943557577072$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $99$ $19031$ $2685087$ $373325059$ $51889333584$ $7212555350435$ $1002544419254169$ $139353667491212755$ $19370159742952286361$ $2692452204199262558486$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{139}$.

Endomorphism algebra over $\F_{139}$
The endomorphism algebra of this simple isogeny class is 4.0.2043717.2.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.139.bp_bat$2$(not in LMFDB)